Let R be a ring with identity, and a,b are elements in R. If ab is a unit, and neither a nor b is a zero divisor, prove a and b are units.
If ab is a unit then (ab)c=1=c(ab) for some c in R.
The Attempt at a Solution
Assume both a and b are not zero divisors, and denote the identity element of R as 1.
Since ab is a unit in R, there exists some c such that (ab)c = 1.
(ab)c=1 [itex]\Rightarrow[/itex] (ca)b = 1 [itex]\Rightarrow[/itex] b is a unit.
Similarly, (ab)c= 1 [itex]\Rightarrow[/itex] a(bc) [itex]\Rightarrow[/itex] a is a unit.
Does the above sufficiently prove the claim in the problem statement? I feel like I'm missing something.